8 COMPLETE HUBERT ALGEBRAS 367 where m is the smallest integer such that 1^ is not isomorphic to any of the ideals Il5 . . . , \k . (If all the 1^ are isomorphic to one or other of Il9 . . . , lk, the induction stops at lk .) Let J be the sequence of indices k so obtained, and for each k e J let lj be the sequence of integers n such that 1^ is isomorphic to Ifc . We define ak to be the Hilbert sum of the 1^ for n e lk . Clearly A is the Hilbert sum of the left ideals ak (6.4.2). Let I be any minimal left ideal in A. Then I must be isomorphic to one of the Ifc, for otherwise it would be orthogonal to all the \'n (1 5.8.1 2(iii)) and hence orthogonal to A itself, which is absurd. The same argument shows that I is orthogonal to all the ah with h ^ k. Hence, as ak is the orthogonal supple- ment of the Hilbert sum of the ah such that h ^ k, we must have I c ak . From this it follows already that ak is the closure of the sum of all the minimal left ideals of A which are isomorphic to l^, and therefore ak is independent of the decomposition of A as the Hilbert sum of the 1^, from which we started. Moreover, for each x e A and each n e Ifc, Vnx is a left ideal which is either {0} or isomorphic to 1^ (15.8.12(iv)), hence is contained in afc. This proves that ak is a two-sided ideal. If 1^ = Ae'n , where e'n is an irreducible self-adjoint idempotent, then V* = e'nA, hence a* = ak. Let I" be a minimal left ideal of the Hilbert algebra ak . We have I" = ake", where e" is a self-adjoint idempotent (15.8.7), and e'ne" cannot vanish for all n e lk , otherwise I" would be orthogonal to all the 1^ (n e lk) and therefore to the closure of their sum, namely to ak : which is absurd because I" ^ {0}, Hence there exists at least one index n E lk such that l'n\'r ^ {0}; since 1^1" is a left ideal in ak , we must have 1^ I" = I", which shows that I" is a minimal left ideal of A, necessarily isomorphic to Vn and therefore to \k (1 5.8.1 2(iii)). If now b is a nonzero two-sided ideal of the algebra ak , it contains at least one minimal left ideal I" of this algebra (15.8.8), hence also contains all the H; (/lei*). But \"\'n = i; (1 5.8.1 2(iii)), and therefore b contains the sum of all the \'n (n e lk). If b is closed, it follows that b = ak. Finally, ah n ak = {0} if h ^ k, because ah and ak are two-sided ideals. A complete Hilbert algebra A is said to be topologically simple if it con- tains no closed two-sided ideal other than A and {0}. It follows from (15.8.13) that the study of the structure of a separable complete Hilbert algebra A reduces completely to the study of the afc, that is to the case where A is topologically simple. (1 5.8.1 4) Let Abe a topologically simple, complete, separable Hilbert algebra. Then for each minimal left ideal I of A, the representation jcf— > U^x) of A in the Hilbert space I is faithful. If A is infinite-dimensional, then so is I. The image of A under Ul is the algebra J$?2(0 of Hitbert-Schmidt operators on I (15.4.8), and there exists ath a, such that b2 =